At Bourne's first arrival to Paris, he invites Marie to his apartment, Marie turns him down at first but later agrees after Bourne insists.
I don't understand why he would do this.
Since the fight (with police in embassy) at Switzerland suggests clearly enough that (to some degree) he's a threat to the public (though he doesn't know why). If he cares about the girl (the only person he knows), he shouldn't bring her into trouble.
Actually, Bourne remains vigilant: When Marie is about to pull her car near the apartment, he suggests she park across the street. He certainly doesn't expect good things to happen.
He speaks French/German and doesn't need a local (BTW Marie is not).
So, why does Bourne invite Marie to his apartment?